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  • View in gallery

    The average diurnal variations of main energy storage terms and the observed net radiation at (a) Daman and (b) Amdo.

  • View in gallery

    Scatterplots for correlation of total energy storage with net radiation (circles) and the simple linear regression function (solid lines) at (a) , (b) , (c) , (d) , (e) , and (f) for Daman.

  • View in gallery

    As in Fig. 2, but for Amdo.

  • View in gallery

    Comparisons of fluxes estimated by NEW1 (solid line), NEW2 (dashed line), and OLD (fine dotted line) with the EC observations (circles) for sensible heat at (a) Daman and (c) Amdo, and for latent heat at (b) Daman and (d) Amdo on three clear days in CP.

  • View in gallery

    Histogram of RMSE of estimated fluxes (W m−2) against the stability intervals on unstable condition for sensible heat at (a) Daman and (b) Amdo and for latent heat at (c) Daman and (d) Amdo by NEW1, NEW2, and OLD in TP.

  • View in gallery

    As in Fig. 5, but for the stable condition.

  • View in gallery

    Comparisons of fluxes estimated by NEW1 (solid line), NEW2 (dashed line), and OLD (fine dotted line) with the EC observations (circles) for (a) sensible heat and (b) latent heat at Daman on three cloudy days in TP.

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A Variational Method for Estimating Surface Turbulent Heat Fluxes with a Consideration of Energy Storage

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  • 1 Key Laboratory for Semi-Arid Climate Change of the Ministry of Education, Key Laboratory of Arid Climatic Changing and Reducing Disaster of Gansu Province, College of Atmospheric Sciences, Lanzhou University, Lanzhou, China
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Abstract

Two approaches are proposed to introduce the surface energy storage into the cost function in a variational method for improving the estimates of surface turbulent heat fluxes. In the first approach, each of the energy storage terms is directly calculated based on available observations, and in the second approach, the total energy storage is fitted by the piecewise linear regression function. The heat flux estimates are validated with the eddy correlation (EC) measurements at two carefully selected stations with different land covers and weather conditions in northwestern China and east of the Tibetan Plateau, respectively. In contrast to the variational method without considering the energy storage in the cost function, two new approaches have improved the heat flux estimates, with the first approach being slightly better, especially around midday and/or under strong unstable conditions. It is also reasonable that the calculated/fitted energy storage with the measurements in the previous time period can be transferred for the heat flux estimates in the later time period. Furthermore, the heat flux estimates with both approaches are less sensitive to the errors in the profiles of temperature, humidity, and wind, as well as energy storage, so they may be more reliable.

Corresponding author address: Shuwen Zhang, College of Atmospheric Sciences, Lanzhou University, Tian-Shui-Nan-Lu 222, Lanzhou 730000, Gansu, China. E-mail: zhangsw@lzu.edu.cn

Abstract

Two approaches are proposed to introduce the surface energy storage into the cost function in a variational method for improving the estimates of surface turbulent heat fluxes. In the first approach, each of the energy storage terms is directly calculated based on available observations, and in the second approach, the total energy storage is fitted by the piecewise linear regression function. The heat flux estimates are validated with the eddy correlation (EC) measurements at two carefully selected stations with different land covers and weather conditions in northwestern China and east of the Tibetan Plateau, respectively. In contrast to the variational method without considering the energy storage in the cost function, two new approaches have improved the heat flux estimates, with the first approach being slightly better, especially around midday and/or under strong unstable conditions. It is also reasonable that the calculated/fitted energy storage with the measurements in the previous time period can be transferred for the heat flux estimates in the later time period. Furthermore, the heat flux estimates with both approaches are less sensitive to the errors in the profiles of temperature, humidity, and wind, as well as energy storage, so they may be more reliable.

Corresponding author address: Shuwen Zhang, College of Atmospheric Sciences, Lanzhou University, Tian-Shui-Nan-Lu 222, Lanzhou 730000, Gansu, China. E-mail: zhangsw@lzu.edu.cn

1. Introduction

Accurate estimates of the surface turbulent heat fluxes have always been the central and practical concern in atmospheric science and hydrometeorology, because the fluxes characterize the mass and energy exchanges between the surface and the atmosphere aloft and have a profound impact on the upper airflow. Besides, accurate estimates of the surface heat fluxes are imperative for accurately validating modeling of surface energy and mass balances (Kalma et al. 2008). The turbulent heat fluxes can be directly measured by an eddy covariance (EC) technique. However, the technique is mathematically complex and requires significant care in setting up and processing data at great expense (Foken and Wichura 1996), so many alternative methods have been proposed for the flux estimates by using other available measurements. For example, based on the Monin–Obukhov (MO) similarity theory, the profile method calculates the turbulent heat fluxes with the measurements of wind velocity, temperature, and humidity at different heights (Panofsky and Dutton 1984); the Bowen ratio energy balance (BREB) method computes the fluxes by using both measurements of vertical gradients of temperature and water vapor pressure and the surface energy budget (Black and McNaughton 1971; Fritschen and Simpson 1989); and the data assimilation technique estimates the fluxes by combining the measurements, for example, remotely sensed land surface temperature, with an additional background information predicted by land surface model (Bateni and Liang 2012; Bateni et al. 2013, 2014; Caparrini et al. 2004).

Because massive observations of near-surface profiles of wind velocity, temperature, humidity, and surface energy budget exist, the method that is able to estimate the surface turbulent heat fluxes only based on these in situ observations may have a good application. Both the profile method and the BREB method can really compute the surface heat fluxes by using these in situ measurements. However, the computed fluxes with the profile method often deviate from the surface energy balance, and the BREB method becomes computationally unstable when the Bowen ratio is in the vicinity of −1. To fully make use of the advantages and to overcome the disadvantages in two conventional methods of profile and BREB, a variational method was introduced for the surface heat flux estimates by Xu and Qiu (1997), and it has been further validated under different weather conditions or over a variety of underlying surfaces, such as forest, water, sea ice, the Gobi Desert, and so on (Cao et al. 2006; Cao and Ma 2005, 2009; Li et al. 2014; Xu et al. 1999; Zhang and Cao 2013; Zhang et al. 2004, 2007; Zhou and Xu 1999). These previous studies showed that the variational method was more reliable and stable than the two conventional methods, with evident improvements in surface heat flux calculations.

To make the heat flux estimates satisfy the surface energy balance in the variational method, the constraint of surface energy balance is introduced into the cost function (i.e., δE = RobGobHλES, where Rob is the observed net radiation; Gob is the observed ground heat flux at the soil–atmosphere interface; H and λE are the sensible and latent heat fluxes to be estimated, respectively; S is the energy storage such as in the canopy; and δE is any residual fluxes associated with errors). But in the field measurements, the soil heat flux plate is generally buried at a certain depth below the surface, so its measurement represents the soil heat flux only at that depth, not the ground heat flux. To make matters worse, a large soil energy storage often exists between the heat flux plate and the soil surface under a few conditions, for example, when the soil is very wet and/or is experiencing freeze–thaw cycles. Furthermore, the measurements of energy storage in the canopy are seldom made. Our primary tests show that not considering these energy storage terms in the variational method will result in a large deviation between the flux estimates and the EC measurements in some situations.

Based on the above considerations, two approaches are proposed to address the energy storage in the variational method. In the first approach, each of the energy storage terms is directly calculated with available measurements, and in the second approach, the total energy storage is fitted by the piecewise linear regression function with a consideration that sufficient observations are not always available for the calculation of each energy storage term in the applications. To evaluate the improvement of the flux estimates with two approaches, the flux estimates are compared with the measurements with the EC method as well as those with the variational method without considering the energy storage. Since the energy storage in summer is largest among the four seasons and the variational method has been evaluated with good results in the other seasons (Cao and Ma 2005, 2009), the two proposed methods are validated only with the summer observations. Furthermore, we will evaluate the possibility of transferring the calculated or fitted energy storage over the previous time period for the heat flux estimates in the later time period (i.e., not calculate/fit the energy storage any longer). Section 2 introduces the measured datasets at two observational sites. The two approaches are described in section 3. The computational results and the sensitivity tests are presented in section 4. The summary and discussion are provided in section 5.

2. Data and weather conditions

The data were collected at two observational stations with completely different land covers. The first station was located in a flat field with soil texture of silt loam at Daman (38°51.331′N, 100°22.334′E; 1556.06 m MSL; Li et al. 2013). The land cover was maize from May to July, with the maximum height approximately 1.8 m. The annual average temperature and precipitation were 7.2°C and 126.7 mm, respectively. The measurements were carried out during the peak summer growing season from 23 June to 10 July 2012, including wind, temperature, and relatively humidity at heights of 3, 5, and 10 m; outgoing and incoming shortwave radiation; and outgoing and incoming longwave radiation. Turbulent fluxes were measured with a three-axis sonic anemometer (CSAT3, Campbell) and an open-path infrared CO2 and H2O analyzer (LI7500, LI-COR) at a height of 4.5 m. The measurements of soil temperature at depths of 0, 2, and 4 cm and soil water content at 2 and 4 cm are used to calculate the soil energy storage. The soil heat flux was measured by the heat flux plate at a depth of 6 cm. The measurements of biomass, leaf temperature, and photosynthetic rate were conducted only on 16, 18, 28, and 29 June 2012 (Liu et al. 2011; Xu et al. 2013).

The second station was located at Amdo (32°14.468′N, 91°37.507′E; ~4700 m MSL) in the east of the Tibetan Plateau from 21 May to 10 June 1998 during an intensive observation period of the Global Energy and Water Cycle Experiment (GEWEX) Asian Monsoon Experiment (GAME). At this site, two very different seasons existed: premonsoon dry season and summer monsoon wet season. In the dry season, the land surface was almost bare, but in the following summer monsoon season, the land surface became covered with short, sparse grasses. In our study, we choose the measurements only in the dry season to prevent the impact of almost daily precipitation on the EC measurements during the summer monsoon season. The near-surface frozen soil began to melt in April and gradually proceeded into deeper layers. On a diurnal cycle, the near-surface frozen soil melted during the daytime and some of the soil water froze again at night during our study period, so a large quantity of energy was needed for this freeze–thaw process, which could not be measured by the heat flux plate at the depth of 10 cm (Tanaka et al. 2001, 2003). The measurements include the horizontal wind speeds at heights of 1.9, 6.0, and 14.1 m; temperature and water vapor pressure at heights of 1.55, 5.65, and 13.75 m; outgoing and incoming fluxes of shortwave and longwave radiation; soil temperature at depths of 0, 5, and 10 cm; soil water content at depths of 4 and 10 cm; and soil heat fluxes at a depth of 10 cm. The EC fluxes for the latent and sensible heat were measured with a three-axis sonic anemometer (DAT-300, Kaijo) and an infrared hygrometer (AH-300, Kaijo) at a height of 3 m (Tanaka et al. 2001, 2003).

The data were collected every 30 min at both stations. The absolute limit test and the abrupt change test have been employed to exclude abnormal observations. The EC data processing includes spike detection, lag correction of H2O/CO2 relative to the vertical wind component, sonic virtual temperature correction, coordinating rotation using the planar fit method, correction for density fluctuation, frequency response correction, etc. In addition, the half-hourly flux data are rejected in the following conditions: (i) when the sensor was malfunctioning, (ii) when precipitation occurred, (iii) when the missing data ratio was larger than 3% in the 30-min raw record, and (iv) when the friction velocity was below 0.1 m s−1 at night (Foken et al. 2004; Liu et al. 2011). After a series of strict quality controls, there are 707 data samples with 205 discarded at Daman, and 683 with 229 rejected at Amdo.

To investigate whether the energy storage calculated/fitted with the measurements over the previous time period can be used or transferred for the flux estimates over the later time period, whole observational time is split into two time periods at each station, and accordingly, the measurements are also divided into two parts. The first/previous time period was on 16, 18, 28, and 29 June 2012 at Daman and from 21 to 26 May 1998 at Amdo, while the second/later time was from 1 to 15 July 2012 at Daman and from 27 May to 9 June 1998 at Amdo.

3. Descriptions of methods

a. Variational method

Based on the MO similarity theory, the profiles of wind, potential temperature, and specific humidity in a horizontally homogenous surface layer can be described by the following equations (Businger et al. 1971):
e1
e2
e3
Here, is the frictional velocity; and are the potential temperature and humidity scale, respectively; is Obukhov length, where g is the acceleration of gravity; zr = zd, where d is the zero-plane displacement; , , and are the roughness lengths for momentum, heat, and humidity, where = 0.0045 m at Amdo (Ma et al. 2002) and = 0.2 m at Daman with an assumption of and (Brutsaert 1982; Lo 1996); and are the potential temperature and humidity at ; k = 0.4 is the Von Kármán constant; and , , and are three stability functions.
For the unstable case , based on Paulson (1970), Dyer and Hicks (1970), and Hicks (1976), and are taken as
e4
and
e5
where . For the stable case , based on Beljaars and Holtslag (1991), and are given by
e6
e7
Here, a = 1, b = 0.667, c = 5, and d = 0.35.
If three parameters of , , and in Eqs. (1)(3) are computed out, the sensible and latent heat fluxes can be estimated as follows:
e8
e9
Here, cp is the specific heat at constant pressure, ρ is the air density, and λ is the latent heat of evaporation.
Considering that the profile formulas are imperfect, all observations are not free of errors, and the number of observations is larger than that of three estimated parameters, the optimal statistical estimation method is a good choice for the heat flux estimates (Sasaki 1970; Lorenc 1986). Considering available measurements and corresponding observational heights, the following form of cost function is defined:
e10
Here, , , and , with , , and calculated by Eqs. (1)(3). The first three terms on the right-hand side of Eq. (10) measure the mismatches between the calculated wind, temperature, and specific humidity in Eqs. (1)(3) and the corresponding observations, respectively; the observations of wind speed and specific humidity are measured directly but the potential temperature is calculated with the help of temperature T and surface pressure by ; and denotes the measurement heights (for details, see section 2). The last term measures the fit to the surface energy balance with , , and S representing the observed net radiation fluxes, observed soil heat fluxes, and the calculated/fitted total energy storage, respectively.

According to the maximum likelihood theory, weights , , , and in Eq. (10) should be inversely proportional to variances of the corresponding items (Lorenc 1986). The observational errors are 0.14 m s−1, 0.2 K, and 4.4 × 10−4 (about 2% relative humidity) for wind, temperature, and humidity at Daman, respectively; errors are 0.5 m s−1, 0.2 K, and 2.2 × 10−4 (about 1% relative humidity) for wind, temperature, and humidity at Amdo, respectively; and the medium-quality class of the EC data indicates that the random error of the sensible heat flux is 10% or 20 W m−2 and the latent heat flux is 15% or 30 W m−2 (Mauder et al. 2006). Accordingly, = 0.14−2 = 50 s2 m−2, = 0.2−2 = 25 K−2, = (4.4 × 10−4)−2 = 5.17 × 106, and = 50−2 = 4.0 × 10−4 W−2 m−4 at Daman; and = 0.5−2 = 4 s2 m−2, = 0.2−2 = 25 K−2, = (2.2 × 10−4)−2 = 2.07 × 107, and = 50−2 = 4.0 × 10−4 W−2 m−4 at Amdo. Although the above weights may not be accurate, fortunately, computations are found not to be sensitive to the weights in the vicinity (within the same orders of magnitudes) of these values (Xu and Qiu 1997).

To obtain optimal estimates of , , and , the gradients of J relative to the three unknowns should be zero:
e11
The quasi-Newton algorithm (Liu and Nocedal 1989) is introduced to minimize the cost function J in Eq. (11). The iterative procedures consist of the following basic steps:
  1. Set the initial guesses: = 0.1 m s−1, = 0.01 K, and = 0.

  2. Calculate J in Eq. (10) and its three gradients by Eqs. (A1)(A3) in the appendix.

  3. If , then the expected estimates are reached. Otherwise, determine a search direction based on the gradients and the search directions of previous iterations.

  4. Determine the search step size and find the minimum of J along the search direction; obtain the new estimates of and return to step 2.

To improve the convergence rate of iteration, should be properly scaled to make the supersurface of the cost function more spherical (Moore 1991; Xu et al. 1994). In this paper, are scaled by 1 m s−1, 0.5 K, and 5 × 10−4, respectively. The minimization procedure is found to converge within no more than 20 iterative steps.

b. First approach: Directly calculating each energy storage term

Because of no direct measurements, the energy storage S during unit time is computed as follows:
e12
where is the soil heat storage between the heat flux plate and land surface, is the canopy heat storage, is the energy used for photosynthesis, and is the energy for other use such as the freeze–thaw process at Amdo.
The soil heat storage is calculated as
e13
where h is the depth of the soil heat flux plate (6 cm at Daman and 10 cm at Amdo), is the volumetric soil heat capacity, and is the soil temperature. Because of limited measurements, Eq. (13) is simplified to
e14
where = 1.0 × 103 Kg m−3 K−1; (=4.2 × 103 J kg−1) is the specific water heat capacity; w is the average soil water content at 2 cm at Daman and 4 cm at Amdo; (=9.0 × 105 J m−3 K−1) is the volumetric heat capacity of dry soil; Δt is the time interval; and was measured at depths of 0, 2, and 4 cm at Daman and 0, 5, and 10 cm at Amdo.
The canopy heat storage at Daman is
e15
where the temperature change is computed over the time interval Δt for the mass of water (kg m−3) and biomass (kg m−3), and and are the specific heat capacities for plant water and biomass, respectively (Meyers and Hollinger 2004).

The estimate of energy storage by photosynthesis at Daman is obtained from the net sum of the energy that is required to break the bonds of the reactants and those in forming glucose and oxygen (the change in the Gibbs free energy). The solar energy is stored in the bond of carbohydrate with about 422 kJ per mol of CO2 fixed by photosynthesis (Wang and Zhang 2011). A canopy assimilation rate of 2.5 mg CO2 m−2 s−1 is equal to an energy flux of 28 W m−2.

Considering the existence of the freeze–thaw process at the near-surface soil layer and almost no plant growth during the studying period at Amdo (Tanaka et al. 2001, 2003), the energy storage mainly comes from and for melting the frozen soil between the heat flux plate and land surface during daytime. The is still calculated by Eq. (14), but it is very difficult to directly calculate (Chen et al. 2012; Niu and Yang 2006). However, considering almost no other sources of energy loss and storage (Tanaka et al. 2001, 2003), we assume that the surface energy residual is all used for melting, that is, . Here, Hob and λEob are the observed sensible and latent heat fluxes, respectively.

For reference, the average diurnal variation of each energy storage term at Daman and Amdo is plotted in Fig. 1. The largest contribution to the total energy storage comes from the soil energy storage, so the main uncertainty in the computed total energy storage is assumed to result from the soil energy storage. According to Eq. (14), the uncertainty in the calculation of soil energy storage is estimated as follows (Box et al. 1978):
e16
The errors in the observations of soil water content Δw, soil temperature , and dry soil heat capacity are 0.02 m3 m−3, 0.1 K, and 5.1 × 104 J kg−1 K−1 at Daman, respectively; and 0.03 m3 m−3, 0.1 K, and 4.1 × 104 J kg−1 K−1 at Amdo, respectively, so the estimated daily averaged uncertainty in the soil energy storage is 10.3 W m−2 at Daman and 12.6 W m−2 at Amdo, which will be used in the sensitivity tests in section 4c. Among the three observational errors, the error in soil temperature is a main source of the uncertainty in the estimation of soil energy storage with a contribution rate of 95.8% at Daman and 96.1% at Amdo.
Fig. 1.
Fig. 1.

The average diurnal variations of main energy storage terms and the observed net radiation at (a) Daman and (b) Amdo.

Citation: Journal of Hydrometeorology 17, 10; 10.1175/JHM-D-15-0193.1

c. Second approach: Estimating the total energy storage with linear regression

Because of needing additional observations in calculating each energy storage term in the first approach, it is infeasible at the conventional observational stations. Therefore, in the second approach, the energy storage as a whole, that is, the total energy storage, is fitted by a piecewise linear regression function. Figure 1 shows that S has a diurnal variation with a quick change around midday and almost no change at night, so for an accurate fitting, the whole day is split into six small intervals, that is, 0000–0600, 0600–0900, 0900–1200, 1200–1500, 1500–1800, 1800–0000 LT (hereinafter referred to as , , , , , and , respectively) and the storage is respectively fitted with the measurements at each interval. Besides, only the measurements on the clear day are used for the fitting.

As shown in Fig. 1, the energy storage S is mainly controlled by the net radiation , so the linear regression function with a form of is supposed at each time interval, where two parameters of a and b are computed by the method of least squares based on the observational data. All the correlations are significant at a confidence level <0.0005. A coefficient of determination, denoted , is adopted to evaluate the accuracy of the fitting function, that is,
e17
where the summation is made for all observations in each specified time interval, is the observed energy storage, and is the average of observed energy storage (Kmenta 1986). The greater the , the higher the accuracy of the fitting function. The regression functions at six time intervals are shown in Figs. 2 and 3, including .
Fig. 2.
Fig. 2.

Scatterplots for correlation of total energy storage with net radiation (circles) and the simple linear regression function (solid lines) at (a) , (b) , (c) , (d) , (e) , and (f) for Daman.

Citation: Journal of Hydrometeorology 17, 10; 10.1175/JHM-D-15-0193.1

Fig. 3.
Fig. 3.

As in Fig. 2, but for Amdo.

Citation: Journal of Hydrometeorology 17, 10; 10.1175/JHM-D-15-0193.1

To compare the heat flux estimates by different methods, both root-mean-square error (RMSE) and percentage of improvement (PI) are used. RMSE is defined as
e18
where the summation is made for M observations and and f are the heat flux observations and estimates, respectively, by three methods, that is, the variational method without considering the energy storage (hereafter OLD), the variational method with the energy storage terms calculated by the first approach (hereafter NEW1), and the variational method with the total energy storage estimated by the second approach (hereafter NEW2). PI measures the improvement of flux estimates by the proposed methods relative to those by OLD, defined as
e19
where RMSE(NEW) is the RMSE estimated by NEW1 or NEW2.

4. Results

The flux estimates are compared with the EC measurements during two different time periods (see section 2 for details). In the first time period [hereafter referred to as the calculation period (CP)], NEW1 uses the directly calculated energy storage terms and NEW2 adopts the fitted energy storage. In the second time period [hereafter referred to as the transferring period (TP)], in NEW1 and at Daman are transferred from those in CP with the other still calculated with the measurements in TP, and in NEW2 the fitted functions at both sites are all transferred from those in CP. The purpose of this experimental setup is to verify the possibility of transferring the calculated or fitted energy storage from a previous to a later time period for the heat flux estimates.

a. Results with calculated/fitted energy storage in CP

First, to have a clear picture of the temporal changes of heat fluxes, the estimates are compared with the EC measurements on three representative sunny days. At Daman, the estimated sensible heat fluxes by NEW1 and NEW2 are generally consistent with the EC measurements, and the problem with the flux overestimates around midday and underestimates at night by OLD is partially resolved (Fig. 4a). Similarly, the latent heat fluxes computed by NEW1 and NEW2 are closer to the EC observations than those by OLD, especially around midday (Fig. 4b).

Fig. 4.
Fig. 4.

Comparisons of fluxes estimated by NEW1 (solid line), NEW2 (dashed line), and OLD (fine dotted line) with the EC observations (circles) for sensible heat at (a) Daman and (c) Amdo, and for latent heat at (b) Daman and (d) Amdo on three clear days in CP.

Citation: Journal of Hydrometeorology 17, 10; 10.1175/JHM-D-15-0193.1

At Amdo, compared with the EC measurements, the heat flux estimates by NEW1 are still the best (especially around midday), those by OLD are the worst, and the performance of NEW2 is between NEW1 and OLD (Figs. 4c,d).

To quantitatively compare the errors of the estimates by three methods, RMSEs of the flux estimates at Daman and Amdo are listed in Tables 14. At Daman, NEW1 produces the smallest RMSE for both sensible and latent heat flux estimates during both daytime and nighttime , NEW2 has slightly larger RMSE than NEW1, and OLD has the largest RMSE at all intervals, although it also has a good performance at night because of small energy storage (Tables 1, 2). The large reductions of RMSEs by NEW1 and NEW2 are centered at and (i.e., 0900–1500 LT) when OLD has large errors. Specifically at and , RMSEs of sensible heat flux estimates by NEW1 have a reduction of 16.8 and 21.3 W m−2 and those of latent heat fluxes have a decrease of 61.1 and 62.3 W m−2 in comparison with OLD, respectively, while at and , the reduction of errors is very small. The pattern of the large reduction of RMSE generally matches well with the large energy storage in Figs. 2c and 2d, showing the importance of considering energy storage terms in improving the flux estimates, especially around midday when the energy storage becomes large. For the whole daytime, RMSEs have a reduction of 14.0 and 12.6 W m−2 for sensible heat fluxes and 51.4 and 46.2 W m−2 for latent heat fluxes, respectively, by NEW1 and NEW2 compared with OLD. For the whole nighttime, RMSEs have a reduction of 6.0 and 4.8 W m−2 for sensible heat fluxes and 7.9 and 6.3 W m−2 for latent heat fluxes, respectively, by NEW1 and NEW2 compared with OLD.

Table 1.

RMSE of computed sensible heat fluxes by NEW1, NEW2, and OLD at six different time intervals at Daman.

Table 1.
Table 2.

As in Table 1, but for latent heat fluxes.

Table 2.
Table 3.

As in Table 1, but for Amdo.

Table 3.
Table 4.

As in Table 3, but for latent heat fluxes.

Table 4.

At Amdo, NEW1 has the best performance with the smallest RMSE during both daytime and nighttime , and NEW2 also has a better performance than OLD (Tables 3, 4). In comparison with OLD, the largest improvement of sensible heat flux estimates with NEW1 appears at and with a reduction of 60.0 and 68.1 W m−2, respectively, and the small reductions of RMSEs by NEW1 and NEW2 are still centered at and (i.e., 1800–0600 LT). For the latent heat fluxes, NEW1 still has the best performance with the smallest RMSE of flux estimates among three methods and NEW2 still has a better performance than OLD; the RMSE reduction of latent heat fluxes is less than that of sensible heat fluxes. For the whole daytime, RMSEs have a reduction of 50.7 and 45.7 W m−2 for sensible heat fluxes and 22.1 and 17.7 W m−2 for latent heat fluxes by NEW1 and NEW2 compared with OLD, respectively. For the whole nighttime, RMSEs have a reduction of 10.4 and 8.4 W m−2 for sensible heat fluxes and 7.8 and 6.2 W m−2 for latent heat fluxes by NEW1 and NEW2 compared with OLD, respectively.

In summary, the larger the errors in OLD, the greater the improvements with both NEW1 and NEW2. Furthermore, introducing the fitted total energy storage into the cost function in NEW2 almost has the same effect as using directly calculated energy storage in NEW1, so NEW2 may have a good application because of no need for additional measurements.

b. Results with the transferred energy storage in TP

RMSEs of the flux estimates at Daman and Amdo at , , , , , and are listed in Tables 14. Similar to the results in CP, NEW1 and NEW2 reduce RMSEs in the heat flux estimates, the large reductions of RMSEs are still at and , and the small reductions are at and .

To compare the relative improvements of heat flux estimates over two time periods (i.e., CP and TP), PI is calculated and listed in Tables 5 and 6. The flux improvements in TP have slight reductions in contrast to those in CP; the larger the PI in CP, the more the reduction of PI in TP. However, the difference of PIs in CP and TP are not very large, with the maximum no more than 5%, showing that it is reasonable to transfer the calculated energy storage terms in NEW1 or the fitted total energy storage in NEW2 in the previous time period to the later time period of the same season for the flux estimates.

Table 5.

Improvement of computed sensible heat fluxes by NEW1 and NEW2 compared to OLD.

Table 5.
Table 6.

As in Table 5, but for latent heat fluxes.

Table 6.

To investigate the distribution of RMSEs with atmospheric stability, the whole range of stability parameter Z/L is divided into 10 nonuniform intervals, where Z is the height at 3 m, with each interval almost containing the same number of observations or estimates for a fair comparison; at Daman the 10 intervals are [−4.00, −0.50), [−0.50, −0.30), [−0.30, −0.12), [−0.12, −0.06), [−0.06, 0.00), [0.00, 3.15), [3.15, 5.23), [5.23, 9.42), [9.42, 15.59), and [15.59, 20.00), and at Amdo they are [−4.00, −0.70), [−0.70, −0.30), [−0.30, −0.10), [−0.10, −0.03), [−0.03, 0.00), [0.00, 3.52), [3.52, 5.83), [5.86, 9.86), [9.86, 15.21), and [15.21, 20.00) (hereafter referred to R1, R2, R3, R4, R5, R6, R7, R8, R9, and R10 for both Daman and Amdo, respectively). Note that R1R5 represent unstable stratification and R6R10 are in stable stratification.

Figure 5 shows that RMSEs by NEW1 or NEW2 have a large reduction when the atmosphere is unstable, especially on the strong unstable conditions such as in R1R3 as compared with those from OLD. On the contrary, when the atmosphere becomes stable, the difference of RMSEs among three methods is small, although two new methods still have smaller errors (Fig. 6). In a word, the more unstable the atmosphere, the greater the improvement of performance with two new methods.

Fig. 5.
Fig. 5.

Histogram of RMSE of estimated fluxes (W m−2) against the stability intervals on unstable condition for sensible heat at (a) Daman and (b) Amdo and for latent heat at (c) Daman and (d) Amdo by NEW1, NEW2, and OLD in TP.

Citation: Journal of Hydrometeorology 17, 10; 10.1175/JHM-D-15-0193.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for the stable condition.

Citation: Journal of Hydrometeorology 17, 10; 10.1175/JHM-D-15-0193.1

c. Further validation

In the following, the performance of two new methods is further evaluated, first on the cloudy days, second by adopting different MO empirical functions, and finally by a sensitivity analysis.

In sections 4a and 4b, the proposed methods have been validated with the observations on the fine weather condition with obvious improvements of the flux estimates. To assess their performance on the cloudy days, the estimates on three representative cloudy days are compared with the EC measurements at Daman (no cloudy days in the observational period at Amdo). Although the sensible heat fluxes have become much smaller under the cloudy sky, two new methods still produce the sensible heat fluxes consistent with the EC measurements and effectively resolve the problem of overestimates around midday by OLD (Fig. 7a). Similarly, the latent heat fluxes computed by NEW1 and NEW2 are closer to the EC observations than those by OLD, especially around midday (Fig. 7b). In essence, two new methods can be used on the cloudy day, although they mainly address the problem of larger energy storage on the sunny day.

Fig. 7.
Fig. 7.

Comparisons of fluxes estimated by NEW1 (solid line), NEW2 (dashed line), and OLD (fine dotted line) with the EC observations (circles) for (a) sensible heat and (b) latent heat at Daman on three cloudy days in TP.

Citation: Journal of Hydrometeorology 17, 10; 10.1175/JHM-D-15-0193.1

Because MO functions are fitted with the measurements at different observational stations with different weather conditions, different forms of functions have been proposed. To evaluate the impact of different functions on the estimates, five additional unstable functions from Businger et al. (1971), Kader and Yaglom (1990), Carl et al. (1973), Högström (1988), and Frenzen and Vogel (1992; hereafter Businger, Kader–Yaglom, Carl, Högström, and Frenzen–Vogel, respectively) and three stable empirical functions from Dyer (1974), Cheng and Brutsaert (2005), and Holtslag and De Bruin (1988; hereafter Dyer, Cheng–Brutsaert, and Holtslag–DeBruin, respectively) are employed. For contrast, the results from the previous studies using the function of Paulson (1970; hereafter Paulson) in unstable condition and the function of Beljaars and Holtslag (1991; hereafter Beljaars–Holtslag) in stable condition in the NEW1 method are also listed in Tables 710. The RMSEs in the flux estimates with different functions have a large variation, and the differences are larger on the unstable conditions (Tables 7, 9) than those on the stable conditions (Tables 8, 10). Surprisingly, no function has a consistently good performance with a smaller RMSE at all intervals at both sites.

Table 7.

RMSE of the heat flux estimates with NEW1 by different similarity functions on unstable condition at Daman.

Table 7.
Table 8.

As in Table 7, but for the stable condition.

Table 8.
Table 9.

As in Table 7, but for Amdo.

Table 9.
Table 10.

As in Table 9, but for the stable condition.

Table 10.

As we know that the calculated surface energy storage is not accurate and the observations of wind, temperature, and humidity also contain errors, a sensitivity analysis is carried out by examining the sensitivities of the heat flux estimates to the errors in the surface energy storage and the profiles of wind, temperature, and humidity. Based on the calculation of the daily averaged uncertainty in the soil energy storage with Eq. (16), artificial errors of ±15 W m−2 are added to the energy storage (see section 3b). Besides, artificial errors of ±0.5 m s−1, ±0.2°C, and ±2.2 × 10−4 are added into the measurements of wind at the highest level, temperature at the lowest level, and specific humidity at the lowest level, respectively. Eight combinations of added errors are listed in Table 11.

Table 11.

Sensitivities of estimated fluxes by NEW1, NEW2, and OLD to the errors in u (m s−1), θ (°C), q (10−4), and S (W m−2) in TP.

Table 11.

Fluxes computed from original data and error-contaminated data are denoted by f and , respectively. The degree of sensitivity is evaluated by the RMS difference (RMSD) defined as
e20
where the summation is made for N time levels (N = 557 at Daman and N = 539 at Amdo in TP). Except for a very slight increase in sensitivity in one case (see last row of Table 11), the heat flux estimates by NEW1 exhibit the lowest degree of sensitivities to the specified observational errors and NEW2 also has a more stable performance than OLD, so two new methods provide more reliable and stable flux estimates than OLD. Since the flux estimates are slightly sensitive to the added errors in the profiles of temperature and humidity, it is suggested that accurate measurements of these sensitive variables be more necessary than the other variables like wind.

5. Summary and discussion

Two approaches are proposed to address the larger errors in the estimates of surface turbulence heat fluxes with the variational method when the surface energy storage is large in some situations. In the first approach (NEW1), each energy storage term is directly calculated, and in the second approach (NEW2), the total energy storage is fitted by the piecewise linear regression function.

Two new methods have been validated with measurements collected at two observational stations (i.e., Daman and Amdo) with different sources of energy storage terms; at Daman, the total energy storage mainly comes from the soil heat storage, canopy heat storage, and energy for photosynthesis, but at Amdo it is mainly from the soil heat storage and the energy needed for melting frozen soil. To verify the possibility of transferring the calculated or fitted energy storage with the measurements over the previous time period for the direct use in the later time period, total observations at each station are deliberately split into two parts corresponding to two different time periods.

A series of tests with the calculated/fitted energy storage at both stations show that NEW1 has the best performance, with the heat flux estimates most consistent with the EC observations, and NEW2 also has an almost equivalent performance with NEW1 by reducing the overestimates of heat fluxes around midday, especially the latent heat fluxes at Daman and the sensible heat fluxes at Amdo. Further studies with the transferred energy storage reach almost similar results, with only slight increases of RMSE but still large reductions of RMSEs around midday or on the strong unstable conditions, showing that the calculated/fitted energy storage with the measurements in the previous time period can be continuously used in the later time period for the heat flux estimates, and thus the applicability of two new approaches is extended.

Sensitivity studies show that NEW1 is the least sensitive to the specified errors in the observations, OLD is the most, and NEW2 is somewhere in between NEW1 and OLD, so NEW1 is the most stable approach for the heat flux estimates. Based on the different degrees of sensitivities, accurate measurements of the temperature and humidity profiles are more important than the wind profile and the calculated energy storage. Besides, different MO similar functions have a relatively large effect on the heat flux estimates under the unstable condition and almost no influence under the stable condition, but none of the selective functions has a consistently good performance at all unstable intervals.

Although two new methods significantly improve the heat flux estimates in comparison with the old one when the energy storage becomes large, especially around the summer midday with clear sky, there are still some deviations from the EC measurements. The possible reasons may be that neither the calculated heat storage terms nor the fitted heat storage are accurate. Besides, the observational errors (including the instruments’ errors and spatial representative errors) and other energy losses will influence the comparative results.

It is assumed that the energy storage is only linearly related to the net radiation, but in fact it is also influenced by the soil physical characteristics, soil water content, land cover, and others. Besides, more influencing factors should be taken into consideration to make the proposed method transfer from one observational site to another different site, not just from one time period to another time period at the same site as in this study.

Acknowledgments

The observational data were kindly provided, respectively, by the GAME-Tibet during the intensive observation period in 1998, and the Heihe watershed allied telemetry experimental research. This work was supported by National Key Basic Research and Development (973) Program of China (2013CB430102 and 2012CB956200), National Natural Science Foundation of China (41575098), and Specialized Research Fund for the Doctoral Program of Higher Education (20120211110019).

APPENDIX

Formulations for the Gradient Components

By using the chain rule of differentiation, the gradient components of the cost function J, defined in Eq. (11) with respect to , are obtained as follows:
ea1
ea2
and
ea3
where
eq1
eq2
eq3
eq4
eq5
eq6
eq7
eq8
eq9
eq10

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